We refer to a prior work [1]} to provide the condition for the evolution of cooperation in general directed networks, with independent structures for interaction and strategy dispersal.
Let \(w_{ij}^{[1]}\)  (resp. \(w_{ij}^{[2]}\) ) denote the edge weight in the interaction (resp. dispersal) network.
Let \(p_{ij}=w_{ji}^{[2]}/\sum _{\ell }w_{\ell i}^{[2]}\)  and \(\pi _i\)  denote the probability that a mutant in node \(i\)  takes over the whole population under neutral drift.
We can obtain \(\pi _i\)  by solving \(\sum _{i}\pi _i=1\)  and \(\pi _i=\sum _{j}p_{ji}\pi _j\) .
Using theorem 1 in Ref. [1]}, we have
\(\eta _{ij} &={\left\lbrace \begin{array}{ll}1+\frac{1}{2}\sum _{k\in \mathcal {N}}p_{ik}\eta _{kj}+\frac{1}{2}\sum _{k\in \mathcal {N}}p_{jk}\eta _{ik} & i\ne j , \\& \\0 & i=j .\end{array}\right.}\) 
